 
import java.text.DecimalFormat;
import java.text.NumberFormat; 

public class spectralnorm
{
	
	private static final NumberFormat formatter = new DecimalFormat("#.000000000");
	
	public static void main(String[] args) {
		int n = 100;
		if (args.length > 0) n = Integer.parseInt(args[0]);
		
		System.out.println(formatter.format(new spectralnorm().Approximate(n)));
	}
	
	private final double Approximate(int n) {
		// create unit vector
		double[] u = new double[n];
		for (int i=0; i<n; i++) u[i] =  1;
		
		// 20 steps of the power method
		double[] v = new double[n];
		for (int i=0; i<n; i++) v[i] = 0;
		
		for (int i=0; i<10; i++) {
			MultiplyAtAv(n,u,v);
			MultiplyAtAv(n,v,u);
		}
		
		// B=AtA         A multiplied by A transposed
		// v.Bv /(v.v)   eigenvalue of v
		double vBv = 0, vv = 0;
		for (int i=0; i<n; i++) {
			vBv += u[i]*v[i];
			vv  += v[i]*v[i];
		}
		
		return Math.sqrt(vBv/vv);
	}
	
	
	/* return element i,j of infinite matrix A */
	private final double A(int i, int j){
		return 1.0/((i+j)*(i+j+1)/2 +i+1);
	}
	
	/* multiply vector v by matrix A */
	private final void MultiplyAv(int n, double[] v, double[] Av){
		for (int i=0; i<n; i++){
			Av[i] = 0;
			for (int j=0; j<n; j++) Av[i] += A(i,j)*v[j];
		}
	}
	
	/* multiply vector v by matrix A transposed */
	private final void MultiplyAtv(int n, double[] v, double[] Atv){
		for (int i=0;i<n;i++){
			Atv[i] = 0;
			for (int j=0; j<n; j++) Atv[i] += A(j,i)*v[j];
		}
	}
	
	/* multiply vector v by matrix A and then by matrix A transposed */
	private final void MultiplyAtAv(int n, double[] v, double[] AtAv){
		double[] u = new double[n];
		MultiplyAv(n,v,u);
		MultiplyAtv(n,u,AtAv);
	}
}

